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Noninvasive control of microwave hyperthermia using retro-focusing

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Title:
Noninvasive control of microwave hyperthermia using retro-focusing
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Sawyer, Marshall Dwayne
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English
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ix, 115 leaves : illustrations ; 29 cm

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Thermotherapy -- Simulation methods ( lcsh )
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bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

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Bibliography:
Includes bibliographical references (leaves 114-115).
General Note:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Electrical Engineering, Department of Computer Science and Engineering.
Statement of Responsibility:
by Marshall Dwayne Sawyer.

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University of Colorado Denver
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Auraria Library
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ocm22839451
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LD1190.E54 1990m .S38 ( lcc )

Full Text
NONINVASIVE CONTROL OF MICROWAVE HYPERTHERMIA
USING RETRO-FOCUSING
by
Marshall Dwayne Sawyer
B.S.E.E., Pennsylvania State University, 1985
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the. degree of.
Master of Science,-
Department of Electrical Engineering and Computer Science
1990


Sawyer, Marshall Dwayne (M.S., Electrical Engineering)
Noninvasive control of Microwave Hyperthermia using
Retro-Focusing
Thesis directed by Professor Jochen Edrich
The general consensus of the researchers dealing in
noninvasive Radiometry and Hyperthermia has been that
multiple antennas used as a phased array or as a correlating
array to focus microwave energy, at depth, in human tissue,
is presently not a viable option. The reasoning is based on
the need for the multiple antennas to be coherently added at
the site of the tumor. This coherence of antenna patterns is
exceedingly difficult when utilizing conventional array
techniques due to the nonhomogeneity and lossiness of the
intervening tissue layers, as well as the lack of a priori
knowledge of the layer thicknesses. A solution to this
problem has been the subject of numerous technical papers
in the recent literature; however, as yet, no appreciable
solutions to the problem have arisen.
This thesis presents a simple and unique solution to
the problem represented by the nonhomogeneous, layered,
and lossy media. The, method allows the various antennas in
the array to add coherently at the site of a neoplasm with
no a priori knowledge of the intervening structure. The
technique developed throughout this thesis uses the
intensity of the incoherent thermal emissions from a tumor


to derive the necessary phase information to focus a phased
array antenna system for the application of hyperthermia.
The methodology of the technique is treated' in detail.
Computer simulation techniques are developed, and
particular test cases are examined and explained. In
particular, this technique showed a focusing accuracy of
less than U5 for focal point location, and a spatial
resolution of less than X/2. for discerning two closely
spaced sources. It proved effective for broad focal regions
necessary to encompass extended or large sources.
Furthermore, the usefulness of this technique was
demonstrated for interstitial array with imprecise antenna
locations.
The form and content of this abstract are approved. I
recommend its publication.
Signed
Jochen Edrich


This thesis for the Master of Science
Degree by
Marshall Dwayne Sawyer
has been approved for the
School of Electrical Engineering and Computer Science
by
Jochen Edrich
Arun K. Majumdar
4/- 30 90
Date


DEDICATION
With love to Joyce for making each day special.


CONTENTS
Figures................................ viii
CHAPTER
1. INTRODUCTION AND OBJECTIVES...............1
2. MEDICAL APPLICATIONS OF HYPERTHERMIA IN
THE TREATMENT OF CANCER..................4
Thermal Emissions from Tumors...........6
Advantages and Limitations of Microwave
Hyperthermia...........................12
3. RETRODIRECTING ARRAYS AND PHASE
INTERFEROMETRY........................ 18
4. RETROFOCUSING ARRAYS.....................24
Homogeneous Lossless Media (Free Space) . 24
Nonhomogeneous Lossless Media (Layered) . 28
Nonhomogeneous Lossy Media.................32
Multiple Sources...........................34
The Applications of Retrofocusing to an
Incoherent Source in Microwave
Hyperthermia...............................36
5. FINITE DIFFERENCE TIME DOMAIN TECHNIQUE
IN ELECTROMAGNETICS..........................39
Formulation of the Difference Equations . . 40


Radiation Boundary Condition...............45
Applications to Retro-Focusing.............47
6. RESULTS.................................... 49
Retro-Focusing Methodology ................50
Homogeneous Lossless Media.................51
Nonhomogeneous Lossless Media..............56
Nonhomogeneous Lossy Media.................63
Specific Cases.............................73
Breast Carcinoma......................73
Interstitial Array Brain Tumor . . 83
7. CONCLUSIONS..................................88
APPENDIX
A. FDTD Code For Personal Computer..............91
REFERENCES ..........................................114
v i i


FIGURES
Figure
2.1. Black Body Radiation for Frequency vs. Intensity. .7
2.2. Blood Perfusion vs. Temperature for Tumor,
Muscle, and Skin.......................................13
2.3. Penetration Depth vs. Frequency.........................16
3.1. Single Baseline Interferometer..........................20
4.1. Single Baseline Interferometer..........................26
4.2. Single Baseline Array with Arbitrary Layered
Media Surrounding the Source...........................30
5.1. Yee Lattice for FDTD....................................42
6.1. Free Space Retro-focusing Layout........................52
6.2. Free Space Source Location..............................53
6.3. Free Space Retro-focusing...............................55
6.4. Dual Sources........................................... 57
6.5. Retro-focusing to Dual Sources......................... 58
6.6. Offset Block and Array..................................60
6.7. Source Radiation from Within Offset Block.............61
6.8. Retro-focusing to Offset Block Source. ... .62


6.9. Lossy Slab with Linear Array............................ 64
6.10. Source Radiation from a Lossy Slab.......................65
6.11. Retro-focusing into a Lossy Slab.........................67
6.12. Source Imbedded in Layered, Lossy Box....................69
6.13. Source Radiation Pattern for Layered Box.................71
6.14. Retro-focusing into Layered, Lossy Box...................72
6.15. Right Chest Cavity Layout................................74
6.16. Extended Source Radiation Pattern
from Chest Cavity...................................... 76
6.17. Retro-focusing to Extended Source
in Chest Cavity.........................................77
6.18. Extended Source in the Chest Cavity
with a Water Bolus......................................80
6.19. Retro-focused Pattern With Water Bolus...................82
6.20. Layout for Interstitial Array and Tumor..................84
6.21. Fields of a Uniformly Excited Offset
Interstitial Array......................................85
6.22. Retro-focusing for the Offset
Interstitial Array......................................87


CHAPTER I
INTRODUCTION AND OBJECTIVES
Attempts to utilize noninvasive phased arrays to
focus energy onto a tumor at depth in human tissue have had
very little success. Hyperthermia phased arrays, in order to
focus their energy in the array near field, require the energy
from each array element to add coherently at the focal point
of interest (onto the cancerous growth). This is a very
difficult task for a conventional phased array system since
the area into which it will focus has unknown physical and
electrical parameters. Specifically the human body is made
up of nonuniform layers of nonhomogeneous, lossy tissue.
This presents a problem since the specific shape of the
layers, their thickness, and the perfusion via blood vessels
can vary substantially over very small regions. For common
phased array processing, this knowledge would necessarily
be known before the proper weighting coefficients can be
applied to the array elements. Determining the weighting
coefficients for the various array elements to achieve some
sort of focus into the human body is therefore nearly
impossible, and would change with heating of the tissue,
movement of the patient, and from treatment to treatment


even if they could be determined. It has therefore been
deemed necessary to develop an alternative method for
determining those coefficients.
This thesis presents a unique approach to the
determination of the necessary weighting coefficients
needed to precisely focus an array onto a tumor. The
technique uses the often higher intensity incoherent
thermal emissions from the tumor to extract the necessary
phase information for the array antennas. The processing
technique is hereafter called "incoherent retrofocusing",
and is the near field equivalent of retrodirecting techniques
used in radio direction-finding. The author will expand upon
the principles of retrodirecting arrays as given in the
literature. The theory for retrofocusing arrays in both
homogeneous, lossless media as well as the more important
nonhomogeneous, lossy media will be developed in extensive
detail since it does not appear in the literature.
Application of this technique as it applies to hyperthermia
treatment of cancer will be discussed, as will numerous
simulated results.
The simulations are done using a Finite Difference
Time Domain (FDTD) program. The FDTD algorithm is two
dimensional in nature, transverse magnetic polarization,
and was designed to run on a personal computer. The FDTD
technique was chosen since it would operate independently
of any assumptions that may be made in the analysis.
2


Numerous simulations were run to determine the
validity of the retrofocusing technique to nonhomogeneous
lossy layers surrounding a thermal source. By using
specific examples, it will be shown that the technique is
efficient, and accurate.
3


CHAPTER 2
MEDICAL APPLICATIONS OF HYPERTHERMIA IN THE
TREATMENT OF CANCER
Heat therapy for various maladies, ranging from the
common cold to the treatment of cancer, has been around
for over 3000 years. Both Homer and Herodotus praised its
therapeutic significance. Historical records tell us of the
use of balneotherapy in ancient Greece, Israel, Egypt,
various portions of Africa, in America among the Indians,
and of course the extensive baths of the Roman Empire [1].
The Romans were the first however to establish a system of
baths consisting of: the sudatorium, for sweating in hot air
or steam; the caldarium, or hot bath; and the cold bath or
frigidarium In recent years studies have shown a lower
incidence of cancer in the breast, testis, penis, and skin
among the populations using hot baths and saunas [2].
The modern medical treatment of cancer by
hyperthermia began in Germany in 1866 when W. Busch
reported a near spontaneous regression of multiple facial
sarcomas in a 43 year old woman following a viral
erysipalas streptococcal infection with fever. Soon after


Busch's discovery and tests, W.B. Coley began experiments
of his own in the U.S. Coley treated cancer by injecting, at
first virulent streptococci, and later toxins derived from
killed organisms, into his cancer patients [3]. Coley had
much success, but was unsure if the fever or the immunal
stimulation was the cause for the regressions.
While Coley and others pursued the treatment by
injection of viruses, other researchers in laboratories were
finding out that cancer cells showed a greater sensitivity
to heat than benign cells, and in fact the combination of
heat with ionizing X-radiation was even more destructive to
cancer cells [4]. With this knowledge, Warren treated 32
terminal cancer patients with hyperthermia, and in 29 of
the 32 cases immediate improvement was noted [5].
The age of hyperthermia treatment of cancer had
emerged. Though the start would prove turbulent, with first
ionizing radiation as a lone treatment, then chemotherapy,
X-ray and chemotherapy combined, and finally in the early
sixties the reemergence of hyperthermia in combination
with X-ray and chemotherapy.
Many issues are still unresolved in hyperthermic
treatment of cancer sych as: early recognition; thermal
dosages; method of treatment, i.e., localized hyperthermia,
regional, or whole body; etc. The next section of this
chapter will deal with the question of recognition.
Specifically, when dealing with microwave thermography
5


and hyperthermia, how and why can a cancerous region be
recognized. In a later chapter this information will be used
to develop a novel scheme for focusing microwave energy
onto a tumor.
Thermal Emissions from Tumors
All tissues of the human body emit radiation across
the electromagnetic spectrum. These emissions are known
as black body radiation, and any object above absolute zero
will radiate electromagnetic energy to an extent governed
by its emittance. Any body, to remain in equilibrium, must
absorb energy at the same rate as it emits. Bodies in this
category emit energy at all frequencies according to Plank's
radiation law. This is shown in figure 2.1, with each of the
major electromagnetic spectra shown. For ease of
calculation, when dealing in the microwave to infrared
region of the curve shown in figure 2.1, Plank's radiation
law can be reduced to the Rayleigh-Jeans approximation for
radiation intensity due to a radiative source, and takes the
form:
2kTf2
Kf.T) e(f) [2.1]
where: l(f,T) = intensity as a function of frequency and
Temperature
k = Boltzmans constant,
6


Brightness Bf, W sr
Figure 2.1. Black Body Radiation for
Frequency vs. Intensity.
7


T = temperature in Kelvin,
f = frequency,
c = speed of light, and
e(f)= emissivity as a function of frequency.
Emissivity is a function of frequency and, in the microwave
region, changes fairly rapidly. Typical values of emissivity
are:
e(f) = .50 at 3 GHz, and
e(f) = .96 at 30 GHz.
As can be seen from the formula, as the temperature
increases, the intensity increases proportionally. Intensity
however is not the measurable quantity of interest in
thermography, we are interested instead in the field
strength due to some temperature T. Field strength, F, and
intensity, I, are related by a proportionality constant A, and
this is given by:
F(f.T) = A [l(f,T)]1/2. [2.2]
Note that an increase in the temperature, T, in equation 2.2
will correspond to an increase in the field strength
proportional to T1/2. This means that a region of greater
temperature produces a greater field strength than an
associated cooler region. This field strength can then be
measured with some precision.
8


With the aforementioned theory on the laws of
radiation, and the knowledge that the intensity due to a
radiative sourpe can be received and measured, it now
remains to be discussed, exactly why a cancer cell can be
differentiated from a normal healthy cell. The answer lies
in the mechanism of cell growth, as does the reasoning for
why cancerous cells are more susceptible to heat than
healthy cells. Therefore, both the generation of higher
thermal emissions by neoplasms, as well as their
susceptibility to hyperthermic treatments will be
addressed in the following discussions.
Gautherie and Albert [6] demonstrated that metabolic
heat production is directly related to the doubling time of a
tumor. That is to say, that tumors with very high growth
rates have a proportionately higher thermal activity level.
Typical breast carcinoma have a doubling time of
approximately 100 days. This is much faster than a typical
healthy cell, and therefore has a much higher thermal
emission than healthy tissue due to its higher temperature.
This means that the previously mentioned field strength in
cancerous tissue will be higher in amplitude than in any
healthy tissue, and therefore cancerous tissue can be
differentiated by a measurement technique based on field
strengths. The method of measurement used for detection
of differential levels of fields is known as radiometry.
Typical radiometers can measure temperature differences
9


as small as 0.1 C. In the clinical world the terms
radiometry and thermometry are used interchangeably,
however radiometry is the proper term since temperature is
not being measured directly. The usage throughout the
remainder of this text shall be thermometry since it is the
recognized term within the industry.
The typical size of tumors found by palpation are from
2 to 2.5 cm in diameter, and those found by mammography
are typically just over 1.0 cm. This means the tumor could
have been in the development stage for as long as eight
years, assuming a constant doubling time of 100 days. As
the tumor grows, the chances of metastasis increase due to
the increased number of neoplastic cells being released by
the mother tumor. Therefore, early detection and
destruction of the tumor is of paramount importance.
Thermometry (radiometry) could provide an easy,
nonionizing, noninvasive procedure for detection of
cancerous cells due to the cells high thermal activity, but
has as yet, not been accepted by large portions of the
medical community.
Now that the mechanism for recognizing a tumor due
to its thermal emissions has been discussed, we will
address the reasons for their destruction from temperature
levels that have little effect on normal cells. Most if not
all malignant cells are more sensitive to heat than are
benign cells in the range of 41-45 C. At about 46 C some


benign cells begin dying, and above 50 C all cells are
destroyed.
The main consideration in the hyperthermic response
of tumors is their vascularity and its relation to blood
perfusion. It has been known for over half a century that
tumor blood flow is restricted at best. This restriction of
the circulation of blood has many compounding effects. The
blood transports oxygen and nutrients into the neoplasm and
carbon dioxide and waste matter out. In addition to these
vital functions, the blood also transfers metabolic heat to
the surface of the skin where it is dissipated into the
environment. As a tumor grows, the blood vessels around
the tumor must grow with it. However, the increased
growth rate associated with malignant neoplasms far
exceeds the development of the surrounding blood vessels.
Furthermore, the blood vessels which are stimulated by the
cancerous cells are in the form of capillaries, and are not
efficient transporters of food and oxygen and its associated
waste products, nor are they efficient for heat transfer.
These capillaries grow and new capillaries are formed, and
the net result is an intricate mesh of capillaries feeding
other capillaries and eventually the capillaries grow back
onto themselves further restricting the perfusion of blood
into the tumor. This effect of the vessels growing back
onto themselves is known as arterio-venous anastomoses,
and is the equivalent of an electrical short circuit.


Therefore, when cancerous cells are heated, due to the poor
circulation in and around the tumor, heat cannot be
transferred out of the tumor fast enough. This causes an
hypoxic condition within the tumor with an associated PH
imbalance due to the buildup of waste materials. In effect
the tumor starves. Meanwhile, the surrounding healthy
tissue survive since the venous structure can easily dilate
and increase the perfusion rate to cool, and feed the healthy
cells. Figure 2.2 shows the rate of perfusion of both
healthy and cancerous tissues as the temperature is
increased. As can be seen, the flow of blood within the
tumor begins decreasing long before any associated
decrease in the healthy tissue.
Advantages and Limitations of Microwave Hyperthermia
Microwave hyperthermia has the distinct advantage of
being able to preferentially heat a small volume of tissue
without appreciably heating the surrounding volumes as
would happen in techniques such as whole body, or regional
hyperthermia. Due to the targetable heating, higher thermal
doses can be used and therefore the treatment period is
shorter. In whole body hyperthermia, the patient is
immersed in a hot bath, or melted paraffin; placed in a hot
water suit which has baffles for circulating the water; or a
technique known as extracorporeal perfusion is used. In any
of the whole body techniques, the highest systemic


zo-wcmjrmj 2 m o z > X o m <-----------------1 > r- m n
TEMPERATURE (CENTIGRADE)
HEATING FOR 30 40 MINUTES
Figure 2.2. Blood Perfusion vs. Temperature for
Tumor, Muscle, and Skin.
1 3


temperature which can be safely applied is approximately
42 C. The treatments must then last for 2 5 hours to
allow the systemic temperature to rise to a high enough
temperature, and for the temperature to be maintained long
enough for a cytotoxic response of the tumor. There are
also numerous complications which can arise with such
treatments; dehydration, systemic shock, hyperpyrexia, and
the limited temperature control due to the slow cooling by
an overloaded circulatory system. Additionally the
extracorporeal perfusion technique requires a major artery
to be severed, the blood routed through a heater and re-
routed back into the artery. None of these techniques afford
the patient any comfort at all. These treatments, due to the
relatively low temperatures used, are necessarily long, and
usually involve a synergistic therapy which is usually a
cytotoxic drug treatment.
In microwave hyperthermia, temperatures as high as
46 C are used, with a typical treatment time of less than
one-half hour. Since the treatment is localized, the
patient's circulatory system is not being overloaded with an
attempt to get rid of excessive heat and therefore, there is
no dehydration. Additionally, with the exception of
interstitial techniques, no surgery is involved in this
treatment. Temperature control is immediate, by simply
turning off the microwave source, and allowing the
circulatory system to cool the region. Since the heating is


very localized, the cooling is extremely rapid. By using an
array, the microwave energy can be focused, producing a
smaller region of more intense heating. This can also be
done with an ellipsoidal dish system as reported by Edrich
[7-8].
There are however some severe drawbacks to
microwave hyperthermia. The most severe limitation is the
depth of penetration of microwaves into fat and muscle
tissue. The human body with its layers of skin, fat, and
muscle represent a lossy layered media to microwave. This
means that the microwave radiation looses energy as it
travels into the tissues. Or to put it more succinctly, the
ability to heat an area decreases as the depth into the
tissue layers increases. Penetration depth versus frequency
is plotted in figure 2.3 for fat, muscle and breast
carcinoma. As can be seen,depth of penetration decreases
with increasing frequency. Complicating this is the
focusing resolution of any array as the frequency is
changed. As the frequency decreases, the wavelength
increases and the effective focusing resolution decreases.
Typically, the best focusing resolution attainable with an
array is on the order of one-half wavelength. These two
facts when coupled together keep microwave hyperthermia
from becoming a totally noninvasive treatment method
since some tumors may require the surgical insertion of
antennas around the tumor.


Depth (cm)
Figure 2.3. Penetration Depth vs. Frequency.
1 6


Of less concern is the thermographic accuracy
necessary for accurate temperature measurements.
Presently, the temperature can be measured accurately, but
without substantial knowledge of the tissue geometry
between the applicators and the tumor, accurate depth
measurements are not realized. Therefore the temperature
distribution in and around the tumor cannot be accurately
ascertained with respect to position. This drawback should
be solved in the future since it is not limited by the
fundamental physical and electrical parameters as is the
case with depth of penetration problem addressed earlier.


CHAPTER 3
RETRODIRECTING ARRAYS AND
PHASE INTERFEROMETRY
Considerable research on both active and passive
retrodirective antenna arrays has been done in the past [9-
12]. Conceptually a retrodirective antenna acts like a
mirror which reflects the incident electromagnetic energy
back to its source. This mirror effect is accomplished, not
by physically moving the array, but by appropriate phase
control on the individual elements of the array.
The principal of this concept is that if a far-away
source transmits a signal which is received by the
retrodirecting array, each of the n-elements in the array
would receive some amplitude and phase according to the
orientation of the incident waveform given as:
[3.1]
with
An = signal amplitude at nth element, and
<])n = signal phase at nth element.
If then the array is fed in a transmitting sense, and
conjugately fed, then the array will transmit back in the


direction of the source. The weighting of the individual
antenna elements would then be:
Ane'jV [3.2]
This concept was originally investigated for use in
satellite communications, and as can be seen by the
formulas above, it required the source to be a coherent
transmitter. A similar technique which actually found its
infancy in WWII was passive direction finding. In general it
was based on the above theory, with the exception that no
retransmission to the source was done. Instead, the source
was located by extracting the direction of arrival from the
incoming coherent waveform's phase information, the n's in
equation 3.1.
Many sources, such as thermal sources and celestial
radio sources are incoherent in nature, or at best they are
partially coherent. Because of this, retrodirection, in the
classical sense as outlined above, could not be used for
source location. Instead, another method which could
translate the incoming incoherent source information into
angle of arrival information was needed. One technique
which does this, and will be explained in detail over the
remainder of this chapter, is the phase interferometer.
A basic phase interferometer is shown in figure 3.1.
It consists of two antennas separated by a distance d. The


INCOMING SIGNAL
Figure 3.1. Single Baseline Interferometer.
20


source is considered to be distant enough that the direct
paths to each antenna from the source are essentially
parallel at the antennas. This is of course a reasonable, and
often used approximation in array theory, which does not
affect the result. The difference in path length between the
source and each antenna is then determined geometrically
as a distance of dcos(), and is depicted in figure 3.1. This
corresponds to a time delay of dcos(<|))/v, where v is the
velocity within the medium (c for the velocity of light).
The received voltages, v1( on antenna 1 and, v2, on antenna 2
can now be written as:
v1 = V1exp[j(tot-pdcos(<()))] [3.3]
and
v2 = V2exp[jcot], [3.4]
where: = magnitude of v1
V2 = magnitude of v2
P = co/c = free space wave number.
At this point we will momentarily depart from the
derivation of the angle of arrival, and discuss the
incoherent source and the physical implications of
incoherency. If a source is completely incoherent, then the
following conditions are true:
21


Radiation from any incoherent source is
statistically independent of all other radiation
sources.
Any differential segment of an extended source, i.e.
a source much larger than a point source, is
statistically independent of all other
differential segments.
Since incoherent sources are statistically
independent of each other, the radiated energy
from any single source can be uniquely
correlated.
With the information given above, we can proceed
with the derivation for angle of arrival from an incoherent
source. Since we assume our source to be incoherent, the
information from that source can be correlated to obtain
the angle of arrival. There are many ways of doing this, but
the one that follows is often shown in the literature [12].
Take the natural logarithm (In) of equations 3.3, and
3.4, and subtract 3.3 from 3.4 to obtain the difference
voltages. Note that the magnitude of v1 and v2 are
essentially equal and will cancel. This gives the equation:
Let A'F be the real part of this difference giving the result:
= jpdcos(<)>).
[3.5]
= pdcos(<))).
[3.6]
22


Thus, the angle of arrival, <|>, of an incoherent far-field
source can be determined using phase interferometry. This
result has been known for some time, and is frequently used
in radio astronomy, and phase monopulse systems. It should
be noted here that in array theory, it has been known for a
long time that a phase distribution between antenna
elements of AY = -pdcos(<|>) will cause the array to radiate
in the direction <|>. This is simply the conjugate angle of
equation 3.6. Though this material is not new,
comprehension is nevertheless essential to the
understanding of the material in the chapters to follow.
23


CHAPTER 4
RETROFOCUSING ARRAYS
The previous chapter, "Retrodirecting Arrays and
Phase Interferomety", was an introduction to the
mathematics and theory which will be used in this chapter.
The material in chapter 3 was not new, nor was the solution
method novel. In this chapter we will investigate array
focusing in the fresnel region of the array for three specific
material cases. To my knowledge, the analysis for layered,
lossless media, and the nonhomogeneous lossy media, has
not been carried out in the open literature or texts, nor has
the analysis for multiple sources. This chapter lays the
groundwork for understanding the significance of this
thesis, with a final section on the applications of
retrofocusing to hyperthermia.
Homogeneous Lossless Media (Free Spaced
The ability of an array to focus on any given region, is
a function of the distance from the array to the focal point,
or more specifically it is a function of the array's focal
distance divided by the arrays length. The term f/d is given


to this ratio and is an outgrowth of optical theory, where f
is the focal distance and d is the lens diameter, or array
length in the case of antennas. This term is actually a bit
nebulous for antenna arrays, since the focal distance can be
changed by differing the phase distribution over the antenna
elements in the array.
For focusing to be effective, the f/d must remain
within the fresnel region of the array. This region extends
from approximately one-wavelength in front of the array, to
a distance of 2d2/?i, where the upper limit is arbitrarily set
by the phase error over the aperture. Focusing in the
Fresnel region of the array will be analyzed from the
retrofocusing perspective throughout this chapter.
In a homogeneous lossless media, which we will
consider to be free space, the concept of retrofocusing is
easily derived. The same two-antenna array configuration
shown in Figure 3.1 and used for retrodirecting arrays will
be used here and is repeated as Figure 4.1, with a few
changes to the receiving geometry with respect to the
source location. Since the energy is received at antenna 2
prior to antenna 1 in our diagram, we will use antenna 2 as
the reference antenna. In this manner the derivation
parallels the chapter 3 analysis. The received signals on
each antenna are given as:
v1 = V^xpUtcot-plR,,-^)]} [4.1]
25


INCOMING SIGNAL
Figure 4.1. Single Baseline Interferometer.
26


v2 = V2exp{jcot}
[4.2]
where: R1 = distance from the source to antenna 1,
R2 = distance from the source to antenna 2, and
all other variables are as defined for
equation 3.3 and 3.4.
As before we make the assumption that the voltages V.,, and
V2 are equal in magnitude due to the very small difference
between distances R1 and R2. Again, as in chapter 3, the
natural logarithm of the two received voltages are
subtracted to get:
ln~" = jcot jcot + jP(R-|-R2) - [4.3]
V1
Allow A1? to be defined as the real part of equation 4.3; the
equation can be rewritten as:
A'F = P(R1-R2). [4.4]
As a simple check of this equation, we let the distance from
the array to the source increase to become a far-field
source. The two paths from the source can again be
considered parallel and the term R1-R2 becomes dcos(<|>).
This is the same result found from chapter 3, equation 3.6.
27


If we now look at the array in the transmitting case,
to obtain a focus at the original location of the incoherent
source we merely need, by reciprocity, to conjugate the
received phase difference A'F and we get:
AY = -p(RrR2). [4.5]
Equation 4.5 offers us the first evidence that retrofocusing
to an incoherent source, in the fresnel region of an array, is
in fact plausible. Its usefulness in free space however,
does not solve the problem of, retrofocusing to an
incoherent source through multiple layers of differing
dielectric constants, varying thicknesses, and losses within
the layers such as the human body represents. As a next
step, we must investigate the effects of dielectric layers
between the incoherent source and the antenna array. In the
next section, the layers will be assumed lossless. The
progression will then be systematic and more easily
understood.
Nonhomogeneous Lossless Media (Layered^
The single most prominent difference between this
case and the one discussed previously is the fact that the
source can take numerous distinct paths to reach each
antenna. This process of partial reflection and partial
transmission through the numerous lossless layers of
varying dielectric constant still produces a loss along the
28


individual ray paths. This must be accounted for in the
analysis.
Figure 4.2 shows the geometry of the same two
element array as in Figure 4.1, with the incoherent source
in the fresnel region, and an arbitrary, lossless, layered
media surrounding the source. Note that two differing path
lengths are drawn to each antenna, and understand that the
total number of paths due to reflection, and diffraction are
numerous. The received voltage at each antenna, during any
time increment, is then the combination of all the various
path lengths from the source to the antenna. For antenna 1
this looks like:
v, = V^expOCcot-pR/)] + V^exp^cot-pR^)] + ...
+ V1nexpD(£ot-pR1n)], [4.6]
where: the voltage on antenna 1 due to path i,
n = total number of paths from the source to
the antenna, and all other constants are
the same as previously defined.
Since the intervening material layers have losses
associated with the partial reflection and transmission of
the energy in each ray, the stipulation that the potentials be
the same for each path length is unreasonable. Equation 4.6
can be approached as have the previous retrofocusing cases;
29


ANTENNA 1 ANTENNA 2
v v
Figure 4.2. Single Baseline Array with Arbitrary
Layered Media Surrounding the Source.
30


that is by first taking the natural logarithm of both sides of
the equation. This gives us:
lnv1 = In V/+ j(a>t-pR11) + In Kcot-pF^2) + ...
+ In V1n+ j(oot-pR1n). [4.7]
Like terms can now be combined. For convenience the
addition of all the path lengths, R/'s, will be termed R.,1.
And in a similar manner the voltage terms will be combined
as V/. Rewriting equation 4.7 then gives:
lnv1 = In V1T+ jncot jpR1T. [4.8]
In the same manner, the term for lnv2 can be ascertained as:
lnv2= In V2t+ jmcot jfJR2T. [4.9]
Subtracting equation 4.8 from 4.9 and low pass filtering to
remove the jmcot and jncot terms gives the familiar ratio of:
where:
V 2 T T
In j =j|3(R1T-R2T),
VI
[4.10]
Mr
-- the ratio of total voltages for all paths
to antenna 2 and antenna 1 respectively.
The magnitudes V2T, and are measurable quantities. The
importance of this is that there is correlation between
31


antennas 1 and 2 for a source in lossless, layered material,
and that the necessary phase coefficients, AT's, can be
ascertained. Taking the real frequency part of equation 4.10
gives us AT, andthe equation is then:
A>F = P(R,t-R2t) = In +ln^ [4.11]
This is the phase coefficient for retrofocusing in an
arbitrary, layered media.
Nonhomoaeneous Lossy Medium
It has been shown in the previous sections, that
regardless of the mediums, layering, or position of the
source, retrofocusing is accomplished. It now remains to be
shown, that in a layered, lossy medium, retrofocusing is
still possible. As has been the case in each of the previous
sections, the analysis will build upon the foundations set
down in the previous sections.
For the lossy, layered dielectric medium investigated
in this section, there are two mechanisms identified as
producing loss. The first is the loss experienced in the
previous section due to the partial transmission and
reflection of the energy through the layered medium. That
loss still exists in this case, and must be accounted for.
The second loss is the attenuation due to the material
conductivity, which was considered to be zero in the
32


previous section. These losses can be combined to form a
total path loss, for each ray path considered.
Beginning with equation 4.6 of the previous section,
given as:
V! = V11exp[j(cot-|3R11)] + V^exptKGot-pR-i2)] + ...
+ VexptKcot-pR^)], [4.12]
it is easily ascertained that the only terms which are
different in value than the ones for equation 4.6 are the
voltage terms v1 and the V/ 's. These terms only differ in
value, not in form, and can therefore be left in the equation
in their present form. Following this reasoning, it then
becomes clear that the retrofocusing coefficients, AY, are
also of the same form as those given for the lossless
layered case. This is shown as equation 4.13 below. It
should be reiterated that although the form is the same, the
values for the terms, v^ and the 's, can be substantially
different.
T T V V2
A A check on equations 4.11, and 4.13 can be made
easily. If the layering and losses of the layers are assumed
to be zero, i.e.: the source and receiving antenna array are
in free space, the result should reduce to the free space
case. For the free space case the values of V.,Tand V2Tare
33


the same. This is because there is only one path from the
source to antenna 1 and one path from the source to antenna
2, and both have the same free space loss. This means that
V2T
the quantity In j is identically 0, and equations 4.11,and
V1
4.13 reduce to the Fresnel region, free space value for AY.
Multiple Sources
The case of retrofocusing to a single source in
layered, lossy media, is of great importance. Of even
greater importance is what happens when there are multiple
sources in the same layered and lossy media. This problem
is a rather simple extension of the previous result for the
single source.
Beginning with equation 4.12, with some minor
changes we have for a two sources problem:
V! = V1 11exp[j(cot-pR1 /)] + V1 21expO(cot-pRi 21)]
+ ^/exptKcot-pR^2)] + V1>22exp[j(cot-pR122)]
+ ... + V1i1mexpU((Dt-pR1i1m)]
+ V12nexp[j(cot-pR1|2n)], [4.14]
where: V1 2n = voltage on antenna 1, due to source
2, and path n, and
R12n= distance to antenna 1, from source
2, and along path n
m number of paths from source 1, and
34


n = number of paths from source 2.
Following the same procedures as outlined for the previous
solutions of the single source, the natural logarithm of the
quantity is taken, and the terms are combined. This yields
the following result:
lnv1 = lnV1 ^ + lnV12T + jmcot + jncot -
jpR, ,T jpR, 2t. [4.15]
The voltage, v2, at antenna 2 due to the multiple sources,
can be ascertained in the same way. This yields the
equation:
lnv2 = lnV2 / + lnV2 2t + jmcot + jncot
' jP^2,1T jPR2,2T-
[4.16]
Subtracting equation 4.15, from equation 4.16, low pass
filtering to remove the mcot and ncot terms, and using only
the real frequency components yields the equation:
(V22T)
.^2 . (V2.1T) . V*2.2 .
In T. + In _ t + A*F,
(V,/)
(V,/)
[4.17]
where:
AP P(R1.1T-R2.1T) + p(R1i2T-R2.2T) [4.18]
This again is the retrofocusing coefficient for a two
antenna array with multiple sources imbedded in a layered,
lossy media. This equation tells us that if the ratio of
35


received voltages due to source 1 are greater than those due
to source 2, a proportional majority of the retrofocused
energy will be redirected to source 1. Also, equation 4.18
shows us that two foci instead of one will be formed due to
the two varying ray path totals from each source. This
result corresponds with the experimental finding of
Schicelstiel [13].
The result given above for AY, is for two sources, but
can easily be extended to three or more sources by the
addition of the obvious terms. This is given as:
A'P-|3(R,i,t-R2i,t)+P(R, 2t-R2i2t)+... +P(R, 2t-R22t). [4.19]
The Applications of Retrofocusino to an Incoherent
Source in Microwave Hyperthermia
One of the major difficulties in hyperthermia
treatment of cancer using arrays is the inability to
determine the proper phasing coefficients to apply to the
antennas in the array. Without these phasing coefficients,
the array cannot be adequately focused onto the tumor, and
may even be focused at the improper location. Another very
important difficulty with array applicators, is how to deal
with the physiological changes of the tissue during heating,
such as varying perfusion rates, expansion of the tissue due
to heating, changes in the material parameters, etc. Though
36


many of these changes are minor, the cumulative effect can
be great when focusing is being attempted onto a small
area. Finally, there is the question of how to deal with
patient movement during treatment sessions that may be
over an hour long. These questions / problems with present
hyperthermia Array Application systems can be solved
conveniently by using the retrofocusing technique presented
in this thesis. This section will explain how this may be
done.
Chapter 2 explained how a tumor acts as a thermal
source of incoherent energy. The sources displayed in the
previous sections of chapter four, which were used for
determining the retrofocusing coefficients, can be
representative of tumors. The layered lossy media, in the
intervening space between the source and antennas,
represent the layers of fat, muscle, skin, and bone in the
human body. It therefore follows that this technique can be
used to retrofocus to a tumor imbedded in the human body,
for the application of hyperthermia.
The specific benefits to be gained from using
retrofocusing, are numerous and shall be addressed
throughout the remainder of this section.
Retrofocusing allows for the determination of the
proper phasing coefficients to focus an array onto the tumor
location. This is done by first using the array in the receive
mode. The tumor acts as a radiating source of incoherent
37


energy. This energy is received and correlated to determine
the retrofocusing coefficients. The correlation can be done
in many different ways, the method outlined throughout
chapter four being one of them. Another method uses simple
time delays in all but one antenna which is used as a
reference. The time delays are varied with respect to the
reference antenna to produce a maximum, or a minimum
when the reference antenna output is added to the other
antenna's output. The time delay corresponding to the
minimum represents the AÂ¥ of the chapter four method. A
maximum would correspond to the conjugate of A^F. Once
the proper phase coefficients are determined, the array can
then be used as a transmitting array, and can then heat the
tissue with a focus at the tumor location. In this manner a
focus can be achieved at the proper location.
By iterating periodically throughout the treatment
period, the antenna system can compensate for patient
movement and varying blood perfusion rates, as well as
changes in the tissue layer's material parameters. Because
of this function, retrofocusing hyperthermia can be a useful
adjunct tool for treatment of cancer.
38


CHAPTER 5
FINITE DIFFERENCE TIME DOMAIN TECHNIQUE
FOR ELECTROMAGNETICS
The Finite Difference Time Domain (FDTD) technique
for electromagnetic field problems was first proposed by
Yee [14]. Yee's paper proposed a computationally efficient
method of solving Maxwell's time dependent curl equations
using finite differences. Yee'S technique samples the
continuous electromagnetic field at distinct points in a
space lattice at distinct and equal time increments. Yee's
algorithm had numerous limitations which until recently
were not solved. Since the field space must be limited at a
finite boundary, spurious nonphysical reflections were
produced at the outer boundary of the space lattice. There
was no means for tracking vector phase information, nor
was there any means for obtaining the unambiguous
sinusoidal steady-state information often needed in
scattering problems. By the mid 1980's these problems had
been solved, or higher order terms were provided to achieve
greater accuracy in the solutions.


Formulation of the Difference Equations
The FDTD method first proposed by Yee, was later
improved upon by many authors. Taflove [15] simplified the
difference equations to reduce the number of actual
multiplications within the algorithm due to the difference
equations themselves. The FDTD algorithm is a discrete
implementation of the following vector equations:
|j +J =VX H [5.1]
dB 3t =VXE- [5.2]
The following system of scalar equations is equivalent to
Maxwell's equations in the rectangular coordinate system.
8FL 1 8EV 8E
8t ~ F ( 8z -;> [5.3a]
8HV 1 8EZ 8Ey
8t ' ~ V ax' 8z ; [5.3b]
8FL 1 6Ev 8EV
8t ( 5y 8x ^ [5.3c]
8E 1 8H7 8HV
8t E <* 8z oEx) [5.3d]
8EV 1 8H* 8HZ
8t : e ( - v 8z 8x oEy) [5.3e]
8EZ 1 8HV 8H
8t s E ( 8x ' 8z aEz) [5.3f]
40


Yee originally introduced a set of finite-difference
equations for the system of equations 5.3a 5.3f. To do
this Yee denoted a space point in a cubic lattice as:
( i,j,k) = (i6,j8,k8) [5.4]
and any function of space and time as:
Fn(i,j,k) = F(iS,j8,k8,n8t),
where 5 = 8x, Sy, 5z is the space increment, and 5t is the
time increment with i, j, k, and n integers. Using centered
finite-difference expressions for the space and time
derivative, a simple to program, second order accurate in 5
expression is derived. These expressions in 8, and in 8t, are
shown in equations 5.5, and 5.6 respectively.
8Fn(i,j,k) FVgjk) Fn(i-^,j,k)
' =-------------1------------+ 0(82)
8x
[5.5]
5Fn(i,j,k)
8t
Fn+1/2(i,j,k) -
Fn~1/2(i,j,k)
8t
+ 0(8t2)
[5.6]
To achieve the accuracy of equation 5.5, and to account for
all the space derivatives in equations 5.3a 5.3f, Yee
positioned the components of E and H about a unit cell of
the lattice as shown in Figure 5.1. To achieve the accuracy
of equation 5.6, Yee proposed to evaluate E, and H at
alternate half time steps. The result of this treatment of
the lattice space, the following system of finite-difference
41


z
i
E
y
Figure 5.1. Vee Lattice for FDTD.
42


equations are formulated from the system of equations 5.3a
- 5.3f:
Hxn + 1/2(i,j+1/2,k+1/2) = Hxn'1/2(i,j + 1/2,k+1/2)
+ |i(i,j+1 /2,k+1 /2)5 t (*.j+1 /2,k+1) Ey (i,j+1/2,k)
+ Ezn(i,j,k+l/2) Ezn(i,j+l,k+l/2) ] [5.7a]
Hyn + 1/2(i + 1/2,j,k+1/2) = Hyn'1/2(i+1/2,j,k+1/2)
+ (j-(i + 1 /2,j,k+1 /2)5 t (i+1.j.k+1/2) Ez (i,j,k+1/2)
+ Exn(i+i/2,j,k) Exn(i+i/2,j(k+i) ] [5.7b]
Hzn + 1/2(i+1/2,j+1/2,k) = Hzn'1/2(i+1/2,j+1/2,k)
+ n(i+1/2pj+1/2,k)8 t Ex (i+1/2,j+1 ,k) Ex (i+1/2,j,k)
+ Eyn(i,j+l/2,k) Eyn(i+1 ,j+l/2,k) ] [5.7c]
Exn + 1(i+1/2,j,k)
G(i+1/2,j,k)8t
1 2e( i+1 /2,j, k)
o(i + 1/2,j,k)5t
' +2e(i + 1 /2,j,k)
Exn(i+1/2,j,k)
_________1________
+ o(i+1/2,j,k)8t
1+2E(i + 1/2,j,k)
5t
e(i+1/2pjpk)8
[Hzn + 1/2(i + 1/2,j+1/2pk)
- Hzn + 1/2(i+1/2,j-1/2pk) + Hyn + 1/2(i+1/2pj,k-1/2)
- Hyn + 1/2(i+1/2pjpk+1/2) ] [5.7d]
43


Eyn+1 ('-j+1/2-k) =
g(i+1/2,j,k)St
1 2e(i + 1 /2,j,k) n .
g(i+1/2,j,k)St Ey ('-J+1/2-k)
2e(i+1/2,j,k)
1+-
+ q(i,j+1/2,k)8t E(i,j+1/2,k)S (i,j+1/2,k+1/2)
1+2E(i,j+1/2,k)
- H n + 1/2(i,j-1/2,k-1/2) + H n + 1/2(i-1/2,j + 1/2,k)
- Hzn + 1/2(i + 1/2,j+1/2,k) ]
[5.7e]
q(i,j,k+1/2)8t
c n+1 /. .. i /0\ _ 2e(i,j,k+1 /2) ^ n /. . i ._\
Ez (i,j,k+1/2) - cy(i,j,k+1 /2)81 Ez (l-J-k+1/2)
1 +2e(i,j,k+1/2)
1
8t
+ q(i,j,k+1/2)8t E(i,j,k+1/2)8 LMy
1+2E(i,j,k+1/2)
[Hvn + 1/2(i + 1/2,j)k+1/2)
- Hyn+1/2(i-1/2,j,k+1/2) + Hxn + 1/2(i,j-1/2pk+1/2)
- Hxn+1/2(i,j+i/2,k+i/2) ] [5.7f]
In the form of equations 5.7a 5.7f, Maxwell's equations
can be implemented on computer. With the above system of
equations, the new value of a field vector component at any
lattice point depends only on its previous value and on the
previous values of the components of the other field vector
at adjacent points.
Numerous simplifications to the above system of
equations can be realized if fixed time steps can be used, as
44


well as fixed space increments. Under those circumstances,
the run time of the algorithm can be reduced.
Since the accuracy of the finite-difference equations
is only second order accurate, the stability of the time and
space increments must be considered. The space
increments should be on the order of Ji/6 or smaller, to
insure the accuracy of the spatial derivatives, as .well as to
model the structure adequately. The spatial coordinates
should be small in comparison to the overall dimensions of
the structure. To insure the stability of the time stepping
algorithm, 8t must be chosen to satisfy the Courant
Stability Criterion, that is, 8t must satisfy the inequality
8t (
1
1 1
2 +o_2
v-1/2
-1
8x 8y Sz
max
[5.8]
where vmax is the maximum wave velocity propogating
within the model [15].
Radiation Boundary Condition
One of the early problems with the FDTD lattice was
dealing with the field vector components at the lattice
boundaries. These values cannot be determined by the
system of finite-difference equations due to the central
differencing of the spatial derivatives. These values must
therefore be computed using an alternative truncation
condition. Since the value of the field vector varies with
45


the angle of incidence to the boundary, no simple exact
solution exists.
For an outgoing scattered-wave field component, Fs,
the variations are in r, 0, and <|>. This can be represented as a
multipole expansion of the scattered field and is given as:
Fs
F0ej(a>t-pr)
A(0,<|>)
B(0,)
,2
[5.9]
Mur [16] formulated a second order approximation based on
the dependence in equation 5.9, to match the outgoing
scattered wave at the boundary of the lattice. For the x = 0
boundary this is given as:
1 52Fs 1 52Fs j_ 52Fs
* c 5x6t c2 St2 + 2 ^ 5y2
52Fs 1
+ ^]}atx=0 = 0 + O(?). [5.10]
The FDTD computation to realize this condition is given (for
the x=0 case) as:
Ezn+1 (o,j,k+i/2) = Ezn'1 (o,j,k+i/2)
c6t-5 n. < n <
+ ^ [E2nt1(i.i,k*,/2) + E2n',(o,i,k.,/2) ]
25
+ ^^[Ezn(-J-k+1/2) + Ezn(l,j,k+1/2) ]
2
+ 25(c5t+5) [ Ezn(o,j+i,k+i/2) 2Ezn(o,j,k+i/2)
46


+ Ezn(o,j-1,k+1/2) + Ezn(l,j+1,k+1/2) 2Ezn(l ,j,k+1/2)
+ Ezn(i,j-i,k+l/2) + Ezn(0,j,k+3/2) 2Ezn(o,j,k+i/2)
+ Ezn(o,j,k-i/2) + Ezn(i,j,k+3/2)
- 2Ezn(i,j,k+i/2) + Ezn(i,j,k-i/2) ] [5.11]
In equation 5.11, Ezn+1(o,j,k+i/2) is the truncation value of Ez
at the point (o,j,k+i/2). The remainder of the boundary
conditions for the Ex and Ey values at the different
truncation planes can be written as well. Note that only the
components tangential to the truncation planes are needed
to completely describe the radiation boundary condition at
each plane.
Applications to Retro-Focusing
The Finite-Difference Time-Domain approach to
solving scattering problems in Electromagnetics has
several unique advantages. The method is a second order
accurate representation of Maxwell's equations in the time
domain. FDTD based results do not reflect any assumptions
made about wave propogation during the analysis of the
retro-focusing technique and should therefore, provide
additional proof of the retro-focusing concept. The FDTD
algorithm implemented in its two dimensional form
provides quick, easy computations of the fields in and
around the scattering structure(s). Additionally it provides
the field values directly and at any increment in time.
47


For the retro-focusing developed in this thesis, the
FDTD algorithm provided a quick check of the technique.
Also, the FDTD algorithm, being a time-domain technique,
allowed the correlation to be done using true time-delays
on the receiving array. This made the implementation of the
retro-focusing technique simple to program, easy to
comprehend, and extremely efficient. As will be shown in
the results chapter of this thesis, the combination of the
FDTD algorithm, coupled with the retro-focusing was
extremely successful. The FDTD algorithm has a spatial
accuracy of less than X/10 as implemented, and therefore
the spatial resolution of the retro-focusing could be
evaluated. In particular the focal region of the array could
be easily discerned in the contour plots of the E-fields.
This will be discussed in further detail in the results
section also.
48


CHAPTER 6
RESULTS
Numerous retro-focusing cases were run during the
course of this study. The methodology behind the incoherent
signal processing used to determine the proper weighting
coefficients for retro-focusing will be outlined. Also in
this chapter, the most important cases will be discussed
and summarized.
Three general retro-focusing results will be shown.
These include the free-space lossless media, the lossless
media with varying dielectrics within the region, and of
course the lossy dielectric media. For each case discussed,
there will first be a source representing the tumor, and an
array in the receiving mode to intercept and process the
incoherent source emissions. The source is then turned off,
the proper weighting coefficients are applied to the array,
and the array transmits a focus at the tumor source
location.
Two specific cases will be presented. The first is for
retro-focusing into a region representing the right chest


region of a woman with a tumor source located in the breast
tissue. Two subcases are included; one with a water bolus
and one without. The second specific case presented in this
chapter is for ah interstitial array located within the
cranial cavity to retro-focus to a brain tumor.
Retro-Focusing Methodology
The Finite-Difference Time-Domain technique was
chosen for the simulation because of its independent nature
from any approximations made within the analysis.
Likewise, when combined with any of the various
correlation schemes, the result still holds independent
proof of the analysis. For the purpose of this thesis, the
best, and also the simplest correlation technique is the true
time delay technique. This was chosen for the processing of
the incoherent source due to its ease of programming.
An incoherent source radiates a time independent
wavefront in all directions. This wavefront has statistical
independence in time as it radiates. Because of this
independent nature, the radiation from any incoherent
source may be correlated. This is done in the FDTD routine
by sensing for a time-averaged maximum (or a null can be
used) at each antenna location. The algorithm is allowed to
run sufficiently long to allow for a steady state condition
to exist in all cell locations of the lattice space. Then the
first antenna to receive a maximum, or null is considered to
50


be zero time, and the number of time steps of the algorithm
are counted for each of the other antennas until they
receive their maximum. This time delay to the other
antennas is the phase differential needed to retro-focus.
This technique worked very well for all test cases run, and
is also the easiest method to implement in the lab, where
the time delay is actually measured by a variable phase
shifter. Results obtained using this method are outlined in
the next sections of this thesis for each of the particular
conditions outlined.
Homogeneous Lossless Media
This is the free-space retro-focusing first discussed
in chapter 4. The source and array are both located in a
homogeneous, lossless (free-space) medium. This is
depicted in Figure 6.1, where the source location is
identified by an 'x', and the array elements are depicted by
o's. The coefficients are determined using the technique
described in the previous section of this chapter.
Figure 6.2 shows the source location and electric field
(Ez) radiation from the source. This is the incoherent
source to which the array will retro-focus. The source is a
uniform line source in the z-direction (out of the paper),
and the fields represent Transverse Magnetic polarization.
The contours for all the cases are located in the x-y plane.
51


o
Figure 6
Source
T X
2X
OlO O O
l^ 52u
6
.1. Free space Retro-focusing Layout.
52


Figure 6.2 Free Space Source Location.
53


Figure 6.3 shows the array in the transmitting mode
of operation and retro-focusing to the source location. The
source has been turned off, but is marked in the figure as an
'x' for clarity. Each of the five array elements are located
by the 'o' along the bottom of the figure. The array elements
are uniform line sources with an applied phase differential
determined by the correlation processing in the receive
mode. The spacing of the array elements was 5X/6. This
was done to show that even in the Fresnel-zone significant
grating lobes can manifest themselves. In the patient, this
will cause hot-spots on and in the healthy tissue regions,
and should be avoided.
As can be seen in the figure, the five element array
was sufficient to retro-focus to the original source
location. Also note that due to the spacing the focusing is
elongated in the y-direction. Closer spacing of the
elements, as well as additional elements in the array,
would compress the y-directed coherency.
Another free space case of particular interest is the
situation of two incoherent sources radiating some distance
apart. For this case an array of eight antennas with XIA
uniform spacing was used. The two sources began with a
separation distance of one-wavelength, and were moved
progressively closer together, until the array could not
discern a difference. It was determined that the coherency
in the y-direction for the re-radiated pattern was very
54


5.00 10.00 15.00 20.00 25.00
Figure 6.3. Free Space Retro-focusing.
-6.2065
55


nearly one-quarter wavelength, although the resolution of
the retro-focusing algorithm was one cell size in the FDTD
Lattice (1/10-wavelength). At approximately X/2 spacing
of the source, the retro-focusing pattern became elongated
to enclose both source locations.
The source locations are shown in Figure 6.4, and the
retro-focused pattern is shown in Figure 6.5. As can be
seen in Figure 6.5, the pattern is grossly elongated with no
distinct focal point, or points within the region. The focal
region is very broad and the energy rolls off relatively
slowly. Also note that due to the spacing of less than X/2,
there are no grating lobes present in the Fresnel-zone.
When the sources approached U4 apart, the retro-
focused pattern had no change even with changes in the
antenna phase coefficients. This is determined to be the
physical limit of the Fresnel-zone coherency of the radiated
pattern. The physical focal point was located between the
two sources, but the focal field encompassed both source
locations.
Nonhomoaeneous Lossless Media
In this example, a region of lossless dielectric will be
inserted around the source location with the retro-focusing
array located outside the dielectric in free space. This
demonstrates the usefulness of retro-focusing in a layered
region.
56


5.00 10.00 15.00 20.00 25.00
*
CONTOUR FROM -20.000 TO 0.000&0E+00 CONTOUR INTERVAL OF 2.0000 PT(3,3)= -22.031
Figure 6.4. Dual Sources.
57


5.00 10.00 15.00 20.00 25.00
CONTOUR FROH -20.000 TO 0.00000E+00 CONTOUR INTERVAL OF 2.0000 PT(3.3>=
Figure 6.5. Retro-focusing to Dual Sources.
-8.1513
58


A single block of dielectric is used to enclose the
source, and separate the source from the array. This is
shown in figure 6.6 where, as before, the source is depicted
as an 'x', and the array elements are depicted as 'o's. The
relative dielectric constant of the dielectric block is 44,
and the surrounding medium is air. The block is off-
centered from the array axis to demonstrate the angular
independence of the retro-focusing. The scale size in
figures 6.7, and 6.8 are 0.5 cm. per unit. The array spacing
for retro-focusing is 2.5 cm. which is slightly less than XI2
in free space. There are 5 receiving/radiating elements in
the array located along the x=2.5 cm. axis.
Figure 6.7 shows the source radiating from within the
dielectric material. The reflections from the impedance
mismatch at the air-dielectric boundary are evident by the
deformation of the fields within the dielectric region. The
radiated energy is received at the array antenna locations,
and after sufficient time has elapsed for the time domain
radiation to reach a steady state condition, the energy is
correlated. The phasing coefficients used to retro-focus
the array are determined by this correlation process. The
source is then turned off, and the antennas are used as a
transmitting array with phase coefficients determined from
the correlation process.
Figure 6.8 shows the array transmitting case. The
proper coefficients have been determined, and the array is
59


---5 cm
~r
2.5 cm
-i#
Figure 6.6. Offset Block and Array.
60


10.00 15.00 20.00 25.00
Figure 6.7. Source Radiation from Within Offset Block.
61


5.00 10.00 15.00 20.00 25.00
Figure 6.8. Retro-focusing to Offset Block Source.
62


focusing onto the original source location. In Figure 6.8, the
array antennas are located at the "o's", and the original
source location is depicted by an "x". As can be readily seen
in this figure, the array easily retro-focuses to the proper
source location.
Nonhomoqeneous Lossy Media
The final case showing the general properties of a
retro-focusing array concerns the layered, lossy dielectric
media. This is the situation which best represents the
human body as a complex scatterer. Two particular
illustrations will be used; the first is a planar slab of lossy
dielectric with a source imbedded in the dielectric, and a
linear array outside in free-space. The second is a block of
lossy dielectric surrounding a source, with a circular array
surrounding the block and located in free-space.
Figure 6.9 shows the geometry for the infinite lossy
dielectric with an embedded source. The relative dielectric
constant of the material is 4.5, conductivity is 0.1, and the
frequency of operation is 2.54 GHz. The retro-focusing
array is located in free space with a spacing between
elements of one-wavelength.
Figure 6.10 illustrates the source radiation pattern.
The scale is .5 cm. per unit. The source is located at the
point x = 14 units and y = 14 units, with the front edge of
the dielectric located along the y = 10 units axis. The
63


(


Figure 6.9. Lossy Slab with Linear Array.
64


CONTOUR FROM -20.000 TO 0.00000E00 CONTOUR INTERVAL OF 2.0000 PT(3,3)= -26.512
Figure 6.10. Source Radiation from a Lossy Slab.
65


retro-focusing array is situated along the y = 4 units axis,
and the individual elements situated at x = 4, 9, 14, 19, and
24. The very distinct boundary region between the
dielectric and free-space can be seen in this figure. This
causes the distortions in the source radiation pattern due to
reflections from the air-dielectric boundary.
The phase coefficients for retro-focusing are
determined in the same manner as described previously, and
the array is allowed to radiate with the source turned off.
Figure 6.11 displays the retro-focusing pattern of the array.
The antenna array locations are easily discerned in the
figure, and the original source location is marked by an "x".
As can be seen in this figure, the array focuses nicely to the
original source location.
Of considerable importance in this example, is the
relative spacing of the antenna elements. The distance
between elements is slightly less than XI2 in free-space.
Within the dielectric however, we must consider the
effective, or apparent spacing of the antennas. For this
array the apparent spacing, due to the higher dielectric
constant of the intervening media, is greater than X/2. This
causes the high grating lobes located on either side of the
focal region in Figure 6.11. These are of great concern when
heating within the human body since we do not want to kill
healthy tissue (tissue not located at the focal point of the
array).
66


5.00 10.00 15.00 20.00 25.00
CONTOUR FROM -30.000 TO 0.00000E00 CONTOUR INTERVAL OF 2.0000
PTI3.31 =
Figure 6.11.
Retro-focusing into a Lossy Slab.
-9.7576
67


To reduce these grating lobes matched antennas at the
air-dielectric interface are needed, or a matching bolus can
be used with the antennas matched to the bolus. In either
case, the physical size of the antenna is reduced by the
dielectric loading necessary to meet this matched
condition. The antennas can then be physically closer and in
effect reduce the grating lobe occurrence. This will be
discussed further in the section titled "Breast Carcinoma".
The next case of interest is the circular array
surrounding a multiple dielectric with an imbedded source
as shown in Figure 6.12. The frequency of operation is 2.54
GHz. Dimensions are given in the figure for the dielectric
blocks, and the source location is centered in the block.
There are four antennas in the array used for retro-
focusing. These are located in Figures 6.13, and 6.14 at:
Antenna 1 X = 4 y = 14,
Antenna 2 X = 9 y = 9,
Antenna 3 X = 9 *< n CO
Antenna 4 X = 14 y = 4,
Antenna 5 X = 14 y = 24,
Antenna 6 X = 19 y = 9,
Antenna 7 X = 19 y = 19,
Antenna 8 X = 24 y = 14.
As in the previous cases the scale of the figures is 0.5 cm.
per unit.
68


o
o
o
Figure 6.12. Source Imbedded in Layered, Lossy Box.
69


Figure 6.13 is the radiation pattern produced by the
source located in the layered material. The phase
coefficients are determined as described in the previous
cases.
Figure 6.14 shows the array retro-focusing to the
original source location. The source position is identified
as a A in the figure, and is located at x= 14, y= 14. The
circular array was chosen over the linear array for this
case, since it would be the most effective and logical
choice for the shape of the material. Examples of this
would be the annular array applicator used for arms, legs,
and necks. For the cylindrical, or nearly cylindrical
structures of small size, linear arrays are not practical
since the antennas farthest from the source location
(tumor) would be at a severe grazing incidence to the tissue
surface. This would cause two problems; the energy would
have a poor match to the interface since no matching
medium such as the water bolus could be practically fitted
between the array and the skin surface, and also the energy
would have farther to travel in a lossy medium and would
loose effectiveness as a heating element of the array. The
circular array configuration allows a water bolus to be
employed between the antenna elements and the skin
surface to improve the match. The circular configuration
also allows the antennas to be more nearly perpendicular to
the tissue air-interface allowing a better match for the
70


5.00 10.00 15.00 20.00 25.00
CONTOUR FROH -20.000 TO 0.00000E*00 CONTOUR INTERVAL OF 2.0000 PT(3.3I= -19.341
Figure 6.13. Source Radiation Pattern for Layered Box.
71


5.00 10.00 15.00 20.00 25.00
Figure 6.14. Retro-focusing into Layered, Lossy Box
14.023
72


fields. For clarity, Figures 6-13, and 6-14 have no water
bolus. Only the principal of retro-focusing for lossy,
layered media was being demonstrated in this section, and
the bolus would have introduced unnecessary complications
to the figure.
Specific Cases
Two specific cases of retro-focusing will be detailed in
this section. The first is a breast carcinoma. For this
example, an extended source approximately 2 cm. X 2 cm. in
dimension will be used for the retro-focusing. The media
will approximate the right chest cavity of a female. The
cases to be shown are with a matching medium simulating a
water bolus between the array and tissue interface. And for
comparison the array without the water bolus will be
shown.
The final case will be a brain tumor simulation. For
this case, an interstitial array of 4 antennas will be used to
retro-focus. This example is simply to demonstrate the
usefulness of the procedure for cases where interstitial
arrays become necessary.
Breast Carcinoma
Figure 6.15 shows the right chest cavity as it was set
up for the retro-focusing. The size was 13 cm across, and 7
cm. deep. The breast itself was modelled as fat with an
73


MUSCLE
* USAGE AS NOTED IN TEXT
Figure 6.15. Right Chest Cavity Layout.
74


underlying muscle layer. The tumor was located in the fatty
tissue of the breast as shown in the figure. The bold outline
marks the location of the water bolus which was used for
certain cases to be described. Although this chest cavity is
not entirely anatomically correct, it does demonstrate
numerous important features such as: a multiple, lossy
dielectric layers of non-uniform shape; a large size
structure; a tumor source of significant size and shape (not
a line source); the effects of a conformal water bolus for
matching; additional antenna capacity due to the water
bolus.
The first case to be studied is the chest cavity
without the water bolus in place. Figure 6.16 shows the
extended source radiation pattern from the chest cavity.
The dotted line in the figure is the tissue-air interface of
the chest. The retro-focusing array for this case is a five
element array along the y = 5 axis at the locations x = 3, 9,
15, 21, and 27. The incoherent source transmission is
received by these elements, correlated in the usual fashion,
and the array then transmits a properly phased signal to
focus at the original source location.
This focusing is shown in Figure 6.17, where the
original source location is outlined by a dash-dot line. As
can be seen the array does an adequate job of retro-focusing
even with an extremely complex layered and shaped medium.
The tissue-air interface is apparent in this figure due to
75


5.00 10.00 15.00 20.00 25.00
5.00 10.00 15.00 20.00 25.
CONTOUR FROM -60.000 TO 0.00000E*00 CONTOUR INTERVAL OF 3.0000 PTC3.31= -35.856
Figure 6.16. Extended Source Radiation Pattern
From Chest Cavity.
76


5.00 10.00 15.00 20.00 25.00
Figure 6.17. Retro-focusing to Extended Source
in Chest Cavity.
-12.704
77


the high level of reflected fields at the interface, as well
as the high attenuation of the signals there. In this example
the field levels drop from -24 dB at the outer edge of the
chest, to -36 dB on the inner side of the interface. This 12
dB loss of field strength corresponds to a 24 dB loss of
effective power for heating. Furthermore the power is
dissipated at the interface of the tissue to air and causes a
large buildup of heat. This bunching of the fields along the
surface of the chest is to be avoided if possible, and is the
purposes of the water bolus.
The field level at the focal point of the array is from
-36 dB to -42 dB from front to back, and -36 dB to -39 dB
from left to right. The front to back variation is due to the
lossiness of the tissue with depth, and the left to right
variation is simply due to roll-off of the fields about the
focal region. These levels will be compared to the levels
attained in the case where a water bolus was inserted.
Water boluses are used in a two-fold manner. First
water is pumped through the bolus to continually cool the
skin, and second, the water bolus acts as an impedance
matching device for the tissue interface further reducing
the amount of power dissipated at the tissue surface. An
added bonus of the water bolus is the high relative
dielectric constant it has in order to achieve a match at the
tissue interface. When the antennas are matched to the
bolus, they can be loaded with dielectric, and therefore
78


their size can be reduced. Since size of the antenna is the
constraint on how closely they can be spaced, this means
that a greater number of antennas can be used in a smaller
area, and thus spread the power needed for heating the
tumor over a larger array. This provides a greater level of
control for the retro-focusing process when dealing with
multiple or extended sources.
Figure 6.18 is the same source described for the
previous case. In this example however, a water bolus has
been added to show its effects. The water bolus shape and
size are shown in Figure 6.15. The location of the antennas
for retro-focusing are:
Antenna 1 X = 2 y = 8
Antenna 2 X = 3 y = 6
Antenna 3 X = 5 y = 5
Antenna 4 X = 7 V = 5
Antenna 5 X = 9 y = 5
Antenna 6 X = 11 y = 5
Antenna 7 X = 13 y = 5
Antenna 8 X = 15 y = 5
Antenna 9 X = 17 y = 6
Antenna 10 X = 19 y = 7
Antenna 11 X = 21 y = 8
These locations correspond to the outer edge of the water
bolus, and the antennas are assumed to be perfectly
79


5.00 10.00 15.00 20.00 25.00
5.00 10.00 15.00 20.00 25.00
CONTOUR FROH -60.000 TO 0.00000E00 CONTOUR INTERVAL OF 3.0000 PT(3.3)=
Figure 6.18. Extended Source in the Chest Cavity
With a Water Bolus.
41 .478
80


matched to the bolus. The tissue-bolus interface is shown
in Figures 6.18, and 6.19 as a dashed line for clarity.
In Figure 6.19 the retro-focused pattern is shown.
The dash-dot line indicates the location of the extended
source. Note that the tissue-bolus interface (dashed-line)
does not show the significant mismatch displayed in the
previous case for the tissue-air interface. Also due to the
better match between the antennas and the tissue provided
by the bolus, the power level at the tumor site is -15 dB to
-24 dB front to back, and -24 dB to -18 dB left to right.
These field levels are significantly higher than those
without the bolus. It should be noted that some increase in
the field levels was due to the additional elements, but this
would account for no more than 6 8 dB of the nearly 20 dB
difference.
Retro-focusing to an extended source in a lossy,
layered environment was successfully demonstrated. The
importance of the water bolus as a matching and cooling
medium was easily discerned from the cases shown. It now
remains to be shown how the retro-focusing responds in an
interstitial environment.
81


5.00 10.00 15.00 20.00 25.00
CONTOUR FROM -60.000 TO 0.00000E00 CONTOUR INTERVAL OF 6.0000 PTI3.31* -31.146
Figure 6.19. Retro-focused Pattern With Water Bolus.
82


Interstitial Array Brain Tumor
Often, interstitial arrays become necessary for
hyperthermia application. The reasons for this are two-
fold. First, in some areas such as the cranial cavity,
resonance conditions can be excited by an external
applicator or array making invasive techniques necessary.
Secondly, and most often, the tumor site is too deep for
non-invasive techniques due to the lossy nature of the
intervening tissue. In either case interstitial array
applicators are the preferred method of hyperthermia
application. This section will concentrate on an interstitial
array of four antennas situated around a tumor. The tumor
size, and antenna locations are given in Figure 6.20.
The tumor being simulated is located in the cranial
cavity. The er for the brain matter surrounding the antenna
tumor er is 55.0 with a conductivity of 1-45 S/m.
equal distances from the tumor center, and from themselves
so they can be excited uniformly to heat the tumor. Since
the array antennas cannot be perfectly located this
simulation looked at what happens when they are slightly
mis-located. Figure 6.21 shows the tumor being excited by
a uniformly excited array of four antennas slightly offset
from the tumor center. The tumor is offset by .2 cm to the
right, and .2 cm to the top. As can be seen in the figure the
energy deposition is centered between the antennas. This
83


Figure 6.20. Layout for Interstitial Array and Tumor.
84


5.00 10.00 15.00 20.00 25.00
ti | -i i
tr
T
T
T
i__I_I__I__I__I_I__L
5.00
CONTOUR FROM -20.000
I t t i i I i i t i I i t i
10.00 15.00 20.00
TO 0.03000E00 CONTOUR INTERVAL OF
x.
I
25.00
2.0000
PTI3.3 I =
Figure 6.21. Fields of a Uniformly Excited Offset
Interstitial Array.
-2.63R1
85


means that the tumor is not being heated efficiently, and
healthy tissue may be adversely affected by the improper
temperature distribution.
Figure 6.22 is the same offset array; however, in this
case, the array is retro-focusing onto the tumor. As can be
seen, the field deposition pattern is not nearly as uniform,
but is centered over the tumor location. Thus the
effectiveness of retro-focusing for interstitial arrays can
be superior to the uniformly excited array if the element
locations are not precise enough.
86


5.00 10.00 15.00 20.00 25.00
CONTOUR FROH -20.000 TO 0.00000E*00 CONTOUR INTERVAL OF 2.0000 PTI3.3)=
Figure 6.22. Retro-focusing for the Offset
Interstitial Array.
-9.6619
87


CHAPTER 7
CONCLUSIONS
This thesis has outlined a method of heating tumorous
tissue within the human body, using an array with no a
priori knowledge of the tissue layering, depth of the tissue
layers, or any of the electrical parameters of the tissue. As
such, it has solved one of the problems previously limiting
the usefulness of focused-phased array hyperthermia
systems.
The analysis outlined in this thesis was progressive,
and used only those approximations necessary to make the
analysis clear and concise. The analysis was verified using
the FDTD algorithm as a separate entity. None of the
approximations made in the original analysis were
duplicated in the FDTD algorithm; therefore, the results
obtained using the FDTD program demonstrate the accuracy
of the retro-focusing technique. This is true regardless of
the presence of multiple, lossy material layers of varying
thickness and shape.
It was shown, in the results section, that retro-
focusing is an accurate and viable technique for very small


(point) sources, as well as larger (extended) sources. The
technique displayed its capability to focus onto two
discrete sources simultaneously with no additional
information concerning the presence of the second source,
or its location. This is a fundamental advantage of the
retro-focusing technique outlined in this paper.
Furthermore, specific cases representing a breast
carcinoma and a brain tumor were considered. The results
for these specific cases displayed the applicability of
retro-focusing under more realistic circumstances.
To further this work specifically, substantial
laboratory studies of real arrays retro-focusing into either
actual tissue or phantoms are needed. This would greatly
enhance the results obtained using the FDTD algorithm.
Also a three-dimensional FDTD program could be developed
for studying a planar array, or any arbitrary three
dimensional array, and it could be used to examine the
effects retro-focusing has on the heating resolution in a
three-dimensional media.
Further areas of general research are needed to
improve array hyperthermia systems. Among the areas
needing considerable work are accurate radiometric
measurements for temperature discrimination versus
volumetric location. Optimization techniques such as
super-resolution could be investigated for improving the
focusing resolution of the arrays. Also studies of various
89


antenna configurations for improved radiative transfer into
the tissue is another area needing a great deal of work.
90


Appendix A
FDTD CODE FOR PERSONAL COMPUTER
This appendix contains the FDTD algorithm detailed in
chapter 5, and used for the results obtained in chapter 6.
The program was designed for implementation on an IBM-PC
with a few minor changes to the algorithm structure to
conform to PC Fortran.